Stationary phase condition: searching for an understanding

Having noted that Isaac Newton made a lot of progress during self-isolation, I've been back to the large Quantum Mechanics text book I was working through last year. Luckily, I kept a diary so one year later I have been able to pick up the threads. I got stuck on the stationary phase condition for deriving the wave packet needed to represent a particle in QM.

 It's the integral above that was bothering me. p stands for momentum and by the De Broglie equation, p=h/lambda. But since lambda is inversely proportional to frequency, then p is another way of choosing different frequencies. Instead of sine or cos, the wave is represented in its more obscure complex exponential form. Integration means adding up. The clue is superposition. So all we are doing is putting waves of different frequencies one on top of the other.

I drew what we'd do for superposition of different frequencies with a L6 class. Three waves are shown, all with different wavelengths and thus different momenta. Notice that they all coincide constructively at x=0 but don't elsewhere. The green curve is only just out with the red curve and just those two added would still give a reasonable tall wave. But the blue waves interferes destructively in a lot of places. So this is my take on the stationary phase condition - in one place and across a narrow range of frequencies, you get constructive interference. Since we are adding an infinite number of frequencies, elsewhere we get total; destructive interference. Hence overall we get a wave packet - a narrowly defined space with waves. It does bother me that it would seem that constructive interference at x = 0 would give infinite amplitude. I have drawn all 3 with the same amplitude but the equation implies that amplitude depends on momentum. So if most momenta have a very small amplitude, perhaps in the limit you get a defined amplitude.

Does concentration affect pH?

When I made my red cabbage indicator, I thought that the pH for bicarbonate of soda was too high so I decided to see if it was because I had made the bicarbonate to strong.

 1 Mole is an amount of chemical that was invented about 200 years ago when atoms were too small to weigh. They reasoned that if you put a large number of atoms together, it would be heavy enough to weigh. Now it turns out that if you get the formula for the chemical and add up all the mass numbers (top numbers on the Periodic Table), then that mass in grams is worth 1 Mole. Next, if you dissolve that in 1000 cubic centimetres of water, the concentration comes out to be 1 Molar. I calculated that 0.84 grams in 100 cubic centimetres (same as millilitres) would be 0.1M.

So I zeroed the scales with the measuring jug on. I noticed that the RESOLUTION of these scales is + or - 1 gram - it measures to the nearest gram so I decided 1g would be close to 0.84g. I kept adding more and more powder and it still said zero.

 Turns out that it won't measure anything less than 10 grams, so that's what I did. I tried to dissolve it in 100 cubic cm of water, but it wouldn't all dissolve so I went for 200 cubic cm. That works out as a concentration of 0.25M. I made 0.125M by doubling the amount of water to 400 cubic cm and stirring well.

 I made 0.0625M by doubling the amount of water to 800 cubic cm. Concentration is INVERSELY PROPORTIONAL to volume of water: double the volume of water, halve the concentration.

 Then I ran out of measuring jug so I poured half into another jug and doubled both to get 0.0313M.

 Look! It doesn't matter how dilute you make it, the pH stays the same.

The reason is that pH depends on how much the molecule can dissociate into H+ and OH- ions. The more dissociation, the stronger the acid or alklai. The less dissociation, the closer to neutral. That is a property of the chemical and is not affected by diluting it.

Making a red cabbage indicator

I took some red cabbage

 I chopped it quite finely and boiled it in a little water.

 Then I crushed it with the potato masher for a minute or two.

 Then I strained it.

When it cooled I used sherry glasses because I don't have any test tubes at home.

I tried it with lemon juice, vinegar, bicarbonate of soda and toothpaste. The little pot on the end is to show the colour of untested red cabbage juice. So acids tend to be pink and alkalis like bicarbonate of soda go green. What I hadn't realised is that the red cabbage is like a universal indicator in that it not only tells you whether it is acid or alkali but also how strong an acid it is. Strength of acid is given as a pH number. The full colour scale is shown here (scroll down when you get there) https://www.compoundchem.com/2017/05/18/red-cabbage/ pH 8 for toothpaste is probably not far off the mark (I'd expect 7.5) , lemon juice as pH 2 is about right, as is vinegar. I used bicarbonate of soda powder so I might have got a different result if I'd watered it down more.

L6 Estimation question A

Here are the electricity cables behind Wigton cemetery. If you enlarge this photograph, you'll be able to see the next pylon in the distance. Here are two questions to answer:
1. Why does the top wire have no insulator when the lower 6 wires hang from large insulators?
2. Estimate some quantities, look other quantities up and use the resistivity equation. Then calculate the TOTAL resistance between 2 pylons.

Rainbows

I have been given these pictures of rainbows spotted from exercise walks in Wigton.

https://www.msn.com/en-gb/lifestyle/lifestyleshoppinghomegarden/childrens-handmade-rainbows-are-popping-up-in-windows-all-over-the-uk-%e2%80%93-heres-why/ar-BB11CWsd?li=AAnZ9Ug&ocid=mailsignout
One of the things that Isaac Newton did whilst he was self-isolating from the plague in 1665 was to experiment with prisms. He shone a narrow beam of sunlight through one prism and got the rainbow spectrum. He was convinced that the colours were present in the light, but he was aware that the prism itself could have been behaving like stained glass. To rule this out, he took the spectrum from one prism and shone it into a second upside down prism. The colours joined together again to make white light. If the prism was like stained glass, that would have been impossible. So in self-isolation, Isaac Newton proved that white light really is made up of the colours of the spectrum. https://micro.magnet.fsu.edu/primer/java/scienceopticsu/newton/ As I understand it, he chose 7 colours because 7 is seen as a propitious number in religion. That's why I struggle to pick out blue from indigo from violet.

GPE with Joe

Here's one of this morning's newspapers! Joe Wicks has been helping me get fit for running for a couple of years now so I was very pleased to see that he has stepped up to be the nation's PE teacher. https://www.youtube.com/channel/UCAxW1XT0iEJo0TYlRfn6rYQ Go on! You know you want to.
Today he was doing jumps that he likened to Pikachu. We think he got at least 30cm off the ground so he emptied his store of chemical energy from his food and filled his store of gravitational potential energy (GPE). GPE = mgh so for a 65kg man, gain in GPE for such a jump = 65 x 9.8 x 0.3 = 191J. The pathway through which one store empties and the other fills is mechanical work. Work done = force x distance so 191 = force x 0.3. The average leg force then becomes 637N. I can do 10 in 18 seconds so 1.8 seconds per jump. Power = work done/ time taken = 191/1.8 = 106 Watts, so basically 100W. Now he had 800000 computers connected so feasibly 2 million people involved. Suppose they were all adult men - that would be 0.2GW. He made some stab at a number like that at the end on Wednesday - the real number is perhaps 0.1GW - so he's not that far off.

How to set up the pendulum experiment

 Tie a small object onto the bottom of the string. It has be heavy enough to make the string go tight. I used a rubber but you could try a small stone or a lump of Blu-Tack.

 Use a ruler and a pen. Measure from the MIDDLE of the object that you tied on. Put a mark at 20cm from the middle of the object, then 25cm, then 30cm, then 35cm etc.

Next hang your pendulum at the 20cm mark. Try to get it so that it doesn't catch too much as it swings. Pull it a couple of cm to one side (10 or 20 degrees - as shown in the photo) and let go.

One whole swing is across AND back, called the TIME PERIOD T. But you are going to time 10 whole swings. We'll call that 10T. The reason we do that is because T is very small and too close to your reaction time. Doing 10T means that there is less RANDOM ERROR in the experiment.